Question: A monkey is swinging from a tree. On each swing, she travels along an arc that is $75\%$ as long as the previous swing's arc. The total length of the arcs from her first $4$ swings is $175\text{ m}$. How long was the monkey's $1^{\text{st}}$ swing? Round your final answer to the nearest meter.
Answer: Notice that the lengths of the monkey's swings form a geometric sequence. The total distance traveled after $ n$ swings is the ${\text{sum}}$ of the first $n$ terms in the sequence. This is called a geometric series. This is the formula for that sum: $ S={a}\left(\dfrac{1-{r}^{ n}}{1-{r}}\right)$ where ${a}$ is the first term and ${r}$ is the common ratio. We can use this formula, along with the given information, to find the length of the $1^{\text{st}}$ swing, $ a$. Using the given information We are given that each swing is ${75\%}$ as long as the previous swing's arc. This is the common ratio $ r$. We are given that the total length of the arcs from her first $ 4$ swings is ${175\text{ m}}$. So the sum of series $ S$ is ${175\text{ m}}$, and the number of terms $ n$ is $ 4$. We are looking for the length of the $1^{\text{st}}$ swing, $ a$. Finding the first term $\begin{aligned} {175}&={a} \cdot \dfrac{1-\left({0.75}\right)^{{4}}}{1-\left({0.75}\right)} \\\\ \dfrac{1-\left({0.75}\right)}{1-\left({0.75}\right)^{{4}}} \cdot {175} &= {a} \\\\ 64\,\text{m} &= {a} \end{aligned}$ Answer The monkey's $1^{\text{st}}$ swing was $64\,\text{m}$ long.